Toward a Curry-Howard Equivalence for Linear, Reversible Computation

Abstract

In this paper, we present a linear and reversible language with inductive and coinductive types, together with a Curry-Howard correspondence with the logic : linear logic extended with least and greatest fixed points allowing inductive and coinductive statements. Linear, reversible computation makes an important sub-class of quantum computation without measurement. In the latter, the notion of purely quantum recursive type is not yet well understood. Moreover, models for reasoning about quantum algorithms only provide complex types for classical datatypes: there are usually no types for purely quantum objects beside tensors of quantum bits. This work is a first step towards understanding purely quantum recursive types.

Categorical Semantics of Reversible Pattern-Matching

“…In this section, we present a small, formal reversible functional programming language in the like of Theseus [12] and its later developments [13,32]. The core feature of Theseus is to define reversible control-flow using pattern-matching.…”

“…If pattern-matching can be added to reversible imperative languages [18], it is particularly relevant in the context of functional languages where it is one of the core construct needed for manipulating structured data. This is for instance emphasized by the several existing languages making use of it [14,15,12,16,13,32]. In the literature, pattern-matching has either been considered in the context of a Set-based semantics [18], or more generally in categorical models making heavy use of rig structures [33] or co-products [25,31] to represent it.…”